(i) According to geometric crystal classes, i.e. according to the crystallographic point group to which a particular space group belongs. There are 10 crystal classes in two dimensions and 32 in three dimensions. They are described and listed in Chapter 3.2
and in column 4 of Table 2.1.1.1. [For arithmetic crystal classes, see Chapter 1.3
and Table 2.1.3.3 in this volume, and Chapter 1.4
of International Tables for Crystallography, Vol. C (2004).]
Crystal family  Symbol^{†}  Crystal system  Crystallographic point groups^{‡}  No. of space groups  Conventional coordinate system  Bravais lattices^{†} 
Restrictions on cell parameters  Parameters to be determined 
One dimension 
– 
– 
– 

2 
None 
a 

Two dimensions 
Oblique 
m 
Oblique 

2 
None 
a, b 
mp 
(monoclinic) 




γ^{§} 

Rectangular 
o 
Rectangular 

7 

a, b 
op 
(orthorhombic) 





oc 
Square 
t 
Square 

3 

a 
tp 
(tetragonal) 






Hexagonal 
h 
Hexagonal 

5 

a 
hp 







Three dimensions 
Triclinic 
a 
Triclinic 

2 
None 

aP 
(anorthic) 






Monoclinic 
m 
Monoclinic 

13 
bunique setting 

mP 





β ^{§} 
mS^{¶} (mC, mA, mI) 




cunique setting 

mP 





γ ^{§} 
mS^{¶} (mA, mB, mI) 
Orthorhombic 
o 
Orthorhombic 

59 

a, b, c 
oP 
oS^{¶} (oC, oA, oB) 
oI 
oF 
Tetragonal 
t 
Tetragonal 

68 

a, c 
tP 






tI 
Hexagonal 
h 
Trigonal 

18 

a, c 
hP 










7 
(rhombohedral axes, primitive cell) (hexagonal axes, triple obverse cell) 
a, α 
hR 


Hexagonal 

27 

a, c 
hP 







Cubic 
c 
Cubic 

36 

a 
cP 






cI 






cF 
^{†}The symbols for crystal families (column 2) and Bravais lattices (column 8) were adopted by the International Union of Crystallography in 1985; cf. de Wolff et al. (1985) .
^{‡}Symbols surrounded by dashed or full lines indicate Laue groups ; full lines indicate Laue groups which are also lattice point symmetries (holohedries ).
^{§}These angles are conventionally taken to be nonacute, i.e. .
^{¶}For the use of the letter S as a new general, settingindependent `centring symbol' for monoclinic and orthorhombic Bravais lattices, see de Wolff et al. (1985) .

(ii) According to crystal families. The term crystal family designates the classification of the 17 plane groups into four categories and of the 230 space groups into six categories, as displayed in column 1 of Table 2.1.1.1. Here all `hexagonal', `trigonal' and `rhombohedral' space groups are contained in one family, the hexagonal crystal family. The `crystal family' thus corresponds to the term `crystal system', as used frequently in the American and Russian literature.
The crystal families are symbolized by the lowercase letters a, m, o, t, h, c, as listed in column 2 of Table 2.1.1.1. If these letters are combined with the appropriate capital letters for the latticecentring types (cf. Table 2.1.1.2), symbols for the 14 Bravais lattices result. These symbols and their occurrence in the crystal families are shown in column 8 of Table 2.1.1.1; mS and oS are the standard settingindependent symbols for the centred monoclinic and the onefacecentred orthorhombic Bravais lattices, cf. de Wolff et al. (1985); symbols between parentheses represent alternative settings of these Bravais lattices.
Symbol  Centring type of cell  Number of lattice points per cell  Coordinates of lattice points within cell 
One dimension 

Primitive 
1 
0 
Two dimensions 
p 
Primitive 
1 
0, 0 
c 
Centred 
2 
0, 0; , 
h^{†} 
Hexagonally centred 
3 
0, 0; , ; , 
Three dimensions 
P 
Primitive 
1 
0, 0, 0 
C 
Cface centred 
2 
0, 0, 0; , , 0 
A 
Aface centred 
2 
0, 0, 0; 0, , 
B 
Bface centred 
2 
0, 0, 0; , 0, 
I 
Body centred 
2 
0, 0, 0; , , 
F 
Allface centred 
4 
0, 0, 0; , , 0; 0, , ; , 0, 
R^{‡} 

3 

1 
0, 0, 0 
H^{§} 
Hexagonally centred 
3 
0, 0, 0; , , 0; , , 0 
^{†}The twodimensional triple hexagonal cell h is an alternative description of the hexagonal plane net, as illustrated in Fig. 1.5.1.8
. It is not used for systematic planegroup description in this volume; it is introduced, however, in the sub and supergroup entries of the planegroup tables of International Tables for Crystallography, Vol. A1 (2010) , abbreviated as IT A1. Planegroup symbols for the h cell are listed in Section 1.5.4
. Transformation matrices are contained in Table 1.5.1.1
.
^{‡}In the spacegroup tables (Chapter 2.3
), as well as in IT (1935) and IT (1952) , the seven rhombohedral R space groups are presented with two descriptions, one based on hexagonal axes (triple cell), one on rhombohedral axes (primitive cell). In the present volume, as well as in IT (1952) and IT A (2002) , the obverse setting of the triple hexagonal cell R is used. Note that in IT (1935) the reverse setting was employed. The two settings are related by a rotation of the hexagonal cell with respect to the rhombohedral lattice around a threefold axis, involving a rotation angle of 60, 180 or 300° ( cf. Fig. 1.5.1.6
). Further details may be found in Section 1.5.4
and Chapter 3.1
. Transformation matrices are contained in Table 1.5.1.1
.
^{§}The triple hexagonal cell H is an alternative description of the hexagonal Bravais lattice, as illustrated in Fig. 1.5.1.8
. It was used for systematic spacegroup description in IT (1935) , but replaced by P in IT (1952) . It is used in the tables of maximal subgroups and minimal supergroups of the space groups in IT A1 (2010) . Spacegroup symbols for the H cell are listed in Section 1.5.4
. Transformation matrices are contained in Table 1.5.1.1
.

(iii) According to crystal systems. This classification collects the plane groups into four categories and the space groups into seven categories. The classifications according to crystal families and crystal systems are the same for two dimensions.
For three dimensions, this applies to the triclinic, monoclinic, orthorhombic, tetragonal and cubic systems. The only complication exists in the hexagonal crystal family, for which several subdivisions into systems have been proposed in the literature. In this volume [as well as in International Tables for Xray Crystallography (1952), hereafter IT (1952), and the subsequent editions of IT], the space groups of the hexagonal crystal family are grouped into two `crystal systems' as follows: all space groups belonging to the five crystal classes 3, , 32, 3m and , i.e. having 3, , or as principal axis, form the trigonal crystal system, irrespective of whether the Bravais lattice is hP or hR; all space groups belonging to the seven crystal classes 6, , 622, 6mm, 2m and , i.e. having 6, , , , , or as principal axis, form the hexagonal crystal system; here the lattice is always hP (cf. Chapter 1.3
). The crystal systems, as defined above, are listed in column 3 of Table 2.1.1.1.
